Bohr model and dimensional scaling analysis of atoms and molecules

Transcription

1 Boh modl and dimnsional scaling analysis of atoms and molculs Atomic and molcula physics goup Faculty: Postdocs: : Studnts: Malan Scully udly Hschbach Siu Chin Godon Chn Anatoly Svidzinsky obt Muawski ui-hua Xi Moochan Kim Kim Utkin Han Xiong higang hang Physics patmnt, Institut fo Quantum Studis, Txas A&M Univsity

2 Outlin: Chmical bond in Boh modl pictu: H, HH, H Intoduction into imnsional scaling analysis: H, H, H Boh modl as a lag- limit of wav mchanics Constaind Boh modl appoach: H, H 3, LiH, B

3 H Potntial ngy (solid cuvs) of H obtaind fom -scaling analysis. ots a th xact ngis.

4 Boh s 93 molcula modl visitd Anatoly A. Svidzinsky, Malan O. Scully and udly. Hschbach PNAS August 3, 005 vol. 0 no H, a.u H H H H , a.u. 0.0 BH () - ( ), a.u B H , a.u.

5 Natu Physics Publishd onlin: 5 August 005 sach Highlights Subjct Catgoy: Quantum physics Atomic and molcula physics 'Boh'n again Andas Tabsing Abstact A look back at Boh's molcula modl offs a fsh pspctiv on th fomation of chmical bonds btwn atoms in hydogn and oth molculs. Although it is possibl to modl th lctonic stuctu of molculs with gat accuacy, such numical mthods povid littl intuitiv insight into lctonlcton intactions. In two paps, in Physical viw Ltts and Pocdings of th National Acadmy of Scincs, Anatoly Svidzinsky and collagus, hav takn a tip down mmoy lan to uncov an intiguing appoach to undstanding th chmical bonds within molculs, and at th sam tim tak a fsh pspctiv on th "old quantum thoy" dvlopd by Nils Boh in 93. Th famous Boh modl intoducd th quantizd natu of lcton obits in on-lcton atoms, long bfo wav mchanics was dvlopd. Much lat, in th 980s, th so-calld -scal appoach povidd a quantitativ dsciption of th two lctons suounding a hlium nuclus, by gnalizing th Schöding quation to dimnsions; th situation lvant to th th-dimnsional wold is dducd by intpolating btwn th and th limits. Howv, nith appoach although ach succssful in its own alm has so fa yildd satisfactoy sults fo twocnt poblms, such as th hydogn molcul. Svidzinsky t al. hav -xamind th -scal appoach and show how a simpl modification can fix its shotcomings. Whas th oiginal did not vn pdict a bound gound stat fo th hydogn molcul, thi nw vsion povids quantitativ valus that a makably clos to thos obtaind fom xtnsiv comput simulations. Futhmo, th authos show that, in th lag- limit, dimnsional scaling can poduc th Boh modl notably by binging in quantum mchanical concpts that w compltly unknown to Boh at th tim. Svidzinsky t al. xplo futh this link btwn 'nw' and 'old', to dmonstat that Boh's plantay modl is indd abl to quantitativly dscib th hydogn molcul and som mo complicatd molculs such as diatomic lithium and givs a cla physical pictu of how a chmical bond foms. fncs. Svidzinsky, A. A., Scully, M. O. & Hschbach,.. Simpl and supisingly accuat appoach to th chmical bond obtaind fom dimnsionality scaling. Phys. v. Ltt. 95, (005). Svidzinsky, A. A., Scully, M. O. & Hschbach,.. Boh's 93 molcula modl visitd. Poc. Natl Acad. Sci. 0, (005)

6 Scinc, Vol 309, Issu 5740, 459, Sptmb 005 ditos' Choic: Highlights of th cnt litatu Chmisty viving Boh Molculs Bfo th Hisnbg-Schöding fomulation of quantum mchanics, th smiclassical Boh- Sommfld thoy succssfully accountd fo quantizd poptis such as th ngy lvls in th hydogn atom. Howv, th focing of closd obits fo paticl motion an afoul of th unctainty pincipl. cntly, th us of scaling, in which th motion of ach paticl is dscibd by a vcto in dimnsions, was usd to intoduc th unctainty pincipl to this ali thoy. Whn poply don, such quations duc to th coct Schöding fom fo 3 but can still b solvd in th mo tactabl limit. This scaling appoach was applid succssfully to atoms but did not yild bound stats fo molculs. Svidzinsky t al. hav dvlopd a scaling dsciption that fully quantizs on of th angls dscibing th intlcton coodinats and poply wights th contibution of lcton-lcton pulsion. Aft application of a lading coction tm in /, th potntial ngy cuvs fo th lowst singlt, tiplt, and xcitd stats of H a in good agmnt with accptd valus aft minimal numical calculation. Th pocdu also yilds asonabl agmnt fo th gound stat of BH. -- (P.. Szuomi) Svidzinsky, A. A., Scully, M. O. & Hschbach,.. Phys. v. Ltt. 95, (005).

7 Boh modl of H molcul a ngy of th systm: p p m m V wh V is th Coulomb potntial ngy. Quantization condition: h m 0 p nh, n,, 3,K unit of lngth (Boh adius) a 0, unit of ngy (Hat) n n V, V a b a b

8 Possibl lcton configuations cospond to xtma of. xtmum quations: Fo n n th a fou xtmum configuations, a.u z 0, 0, i, i i 4 3 Σ u H 3 H >H H >H H (>.68) Boh modl: xact valu: -.0 H -. >H (>.) Σg "xact" valus , a.u..0a.u., B 0.00a.u..4a.u., 4.74V B.73V

9 Chag distibution in H molcul chag dnsity chag dnsity H A B z, a.u. 0.8 a.u. A B - 0 z, a.u..4 a.u., a.u Position of chag pak dnsity z lativ to th cnt of H molcul / Boh modl Hund-Mullikn , a.u.

10 Boh modl of HH Fo N lctons th modl ducs to finding xtma of th ngy N i n i i V (,, K N, ) Fo HH th th lctons cannot occupy th sam lowst lvl of HH. Fo th configuation n n n 3 th ight ngy cosponds to a saddl-point ath thn to a global minimum. ) ) -( ), a.u HH "xact" valus 0.0 Boh modl , a.u

11 Boh modl of H ()-( ), a.u Gound stat H H H "xact" valus V

12 Boh modl of atoms N i n i i V (,, K N ) Symbol Boh a.u. xact a.u., % 3 H Li B B C N O F N

13 Out-shll lctons of Cabon fom a gula ttahdon in Boh modl. This is simila to bond stuctu of mthan CH 4.

14 -scaling analysis of H atom (gound sat) Schöding quation in 3 dimnsions h a0 unit of lngth (Boh adius) m Ψ Ψ -scaling tansfomation L, a 0 Ψ ( ), 4, 4 ( ) unit of (-) / ngy (Hat) Φ, 3 covs 3-dimnsional valus In th scald vaiabls Schöding quation ads ( -) ( ( 3) ) ffctiv potntial Φ Φ

15 Φ Φ ) ( 3) ( ( -) Limit : with th answ in 3 dimnsions coinsids K ( -) 4 c c Th ngy function is minium at ) ( / xpansion 0 xact hydogn ngy in dimnsions ( ) 3 n

16 Φ Φ ) ( 3) ( ( -) / coction Φ Φ Kp tms contibuting in / ~ Hamonic oscillato Na th minimum Φ Φ 4 ~ ~ Shift in th ffctiv potntial / coction 0

17 Convntional Convntional -scaling analysis of scaling analysis of H H atom atom, 4 ) ( 5.7%.9037 a.u.,.7377 a.u., XACT ) ( 4 θ H cos ˆ H θ sin sin sin L θ θ θ θ ( ) ( ), sin,, Jacobian θ θ J, / Φ Ψ J ( ) θ θ θ cos sin,, Ө -scaling tansfomation Hamiltonian of H atom limit: o a.u., θ

18 / xpansion ( ) L Hschbach, J. Chm. Phys. 84, 838 (986) Wittn Phys. Today 33, 38 (980) L 9 Accuacy with spct to xact sult Hschbach -5.7% / 0.96% / -.0% / 3.8% 40 / 4 -.0% L Wittn -58.% -7.% -.% -4.% -3.% Goodson, Lopz-Caba, Hschbach & Mogan [J. Chm. Phys. 97, 848 (99)] calculatd th xpansion cofficints to od 30 by a cusiv pocdu. Analysis of th cofficints lucidats th singulaity stuctu in th limit which xhibits aspcts of a squa-oot banch point. Using Pad-Bol summation thy obtaind 9 significant figus fo th H gound stat ngy.

19 H molcul molcul φ z ( ) Φ Ψ ( )/ ( ) ( ) ( ) ( ),z,,z,, 4 4 ( ) φ, z z V,,,, Ĥ φ φ φ φ z 3 3 sin sin V φ sin Convntional Convntional -Scaling Scaling V Boh modl Boh modl -scaling scaling ( ) Φ Ψ ) ( sin 3) / ( )/ ( ϕ limit: z z

20 Compaison of diffnt -scalings fo H, a.u Convntional -scaling () () -. Boh 3 (xact) , a.u. V sin φ V Gound stat () of H molcul in th limit calculatd in two -scaling schms (solid lins) and th ``xact ngy in th dimnsions (dots).

21 / coction fo H molcul z z V ff V (,, z, z, φ, ), a.u H -scaling analysis Boh modl ( ) 8 V ff, a.u a.u including / coction. a.u..6 a.u , a.u. z -z ~ K Φ A Φ K ~ A s z s z

22 Modifid / coction, a.u Boh modl including modifid ( ) fist / coction H -scaling analysis, a.u., a.u /, a.u Including / coction -.0 intpolation by -.5 3d od polynomial Boh modl , a.u., a.u. Gound stat () of H molcul in th limit and including modifid / coction Intpolatd -scaling () of H

23 Constaind Boh modl, a.u H , a.u. Molcula axis quantization (cuv ) Atomic quantization (cuv ) Solution xists at >.77 a.u. V V a b

24 H b A B In quantum mchanics th lcton is a cloud with chaactistic siz. Intaction potntial btwn th cloud and th nuclus B is Φ(,). In th Boh modl w tat lcton as a point paticl locatd distanc fom A. Position of th point lcton on th sph givs ight quantum mchanical answ fo th paticl intaction with th nuclus B if b Φ(, )

25 ffctiv intaction potntial btwn lcton and th opposit nuclus H molcul Φ Ψ Ψ b Hitl-London tial function Ψ a ( ) b() ± b() a() a a b b Φ α π 3 α a a() ± S a Aft intgation Φ(, ) ± S () b α π 3 α b b() d Coulomb intgal / ± S a() b() ± S b xchang intgal / α d,, S a( ) b() d / S 3

26 Constaind Boh modl of H ngy function Constaints a b a b V a b Φ( b Φ( a, ), -0.5, ), H a b -0.6, a.u Hitl-London (ffctiv chag) Constaind Boh modl (HL) Σu B 4.50 V Σg Bxact 4.74 V , a.u.

27 Hitl-London vs Hund-Mullikn (singlt) ffctiv potntial Hitl-London Ψ a ( ) b() b() a(), Φ HL (, ) S / S / Hund-Mullikn Mullikn Ψ ( a ( ) b() )( a() b() ), Φ HM (, ) S / /, a.u H Constaind Boh modl (HL) Constaind Boh modl (HM) , a.u. B 4.50 V Bxact 4.74 V B 4.99 V

28 Gound stat of H 3 molcul Constaind Boh modl appoach H 3 H H opposit spins ngy function Constaint V 3, Φ (, ) s Φ t (,) V Coulomb potntial ngy

29 (), a.u. -.5 lina H , a.u. ngy function V, V Coulomb potntial ngy 4 H 3 3 Constaint Φ 4 H (, ) s Φ t (,) H

30 (), a.u. ngy function V, H H , a.u. tiangl H lcton lcton H Constaints 4 Φ s (, ) Φ (, ) s 3 Φ t (, ) 3

31 Gnalization to oth molculs Constaint quation a N [ Φ (, ) Φ (, ) K Φ (, )] N

32 () - ( ), a.u Li LiH H , a.u. Constaint [ Φ (, ) (, ) ] s Φ t

33 ()-( ), a.u B B B ngy function n V, wh n , a.u. 4 Constaint [ 3Φ (, ) (, ) ] s Φ t